«Contents 1. Introduction 1 2. Some properties of abelian schemes and modular curves 3 3. The Serre-Tate theorem and the Grothendieck existence ...»
1. Introduction 1
2. Some properties of abelian schemes and modular curves 3
3. The Serre-Tate theorem and the Grothendieck existence theorem 7
4. Computing naive intersection numbers 10
5. Intersection formula via Hom groups 12
6. Supersingular cases with rA (m) = 0 15
7. Application of a construction of Serre 21
8. Intersection theory via meromorphic tensors 29
9. Self-intersection formula and application to global height pairings 34
10. Quaternionic explications 43 Appendix A. Elimination of quaternionic sums (by W.R. Mann) 49 References 64
1. Introduction The aim of this paper is to rework [GZ, Ch III] based on more systematic deformation-theoretic methods so as to treat all imaginary quadratic ﬁelds, all residue characteristics, and all j-invariants on an equal footing.
This leads to more conceptual arguments in several places and interpretations for some quantities which appear to otherwise arise out of thin air in [GZ, Ch III]. For example, the sum in [GZ, Ch III, Lemma 8.2] arises for us in (9.6), where it is given a deformation-theoretic meaning. Provided the analytic results in [GZ] are proven for even discriminants, the main results in [GZ] would be valid without parity restriction on the discriminant of the imaginary quadratic ﬁeld. Our order of development of the basic results follows [GZ, Ch III], but the methods of proof are usually quite diﬀerent, making much less use of the “numerology” of modular curves.
Here is a summary of the contents. In §2 we consider some background issues related to maps among elliptic curves over various bases and horizontal divisors on relative curves over a discrete valuation ring.
In §3 we provide a brief survey of the Serre-Tate theorem and the Grothendieck existence theorem, since these formthe backbone of the deformation-theoretic methods which underlie all subsequent arguments.
Due to reasons of space, some topics (such as intersection theory on arithmetic curves, Gross’ paper [Gr1] on quasi-canonical liftings, and p-divisible groups) are not reviewed but are used freely where needed.
In §4 we compute an elementary intersection number on a modular curve in terms of cardinalities of isomorphism groups between inﬁnitesimal deformations. This serves as both a warm-up to and key ingredient in §5–§6, where we use cardinalities of Hom groups between inﬁnitesimal deformations to give a formula (in Theorem 5.1) for a local intersection number (x.Tm (xσ ))v, where x ∈ X0 (N )(H) is a Heegner point with CM by the ring of integers of an imaginary quadratic ﬁeld K (with Hilbert class ﬁeld H), σ ∈ Gal(H/K) is an element which corresponds to an ideal class A in K under the Artin isomorphism, and Tm is a Hecke Date: February 11, 2003.
This work was partially supported by the NSF and the Alfred P. Sloan Foundation. I would like to thank B. Gross and J-P.
Serre for helpful suggestions, and Mike Roth for helping to resolve some serious confusion.
2 BRIAN CONRADcorrespondence with m ≥ 1 relatively prime to N. An essential hypothesis in Theorem 5.1 is the vanishing of the number rA (m) of integral ideals of norm m in the ideal class A. This corresponds to the requirement that the divisors x and Tm (xσ ) on X0 (N ) are disjoint. Retaining the assumption rA (m) = 0, in §7 we develop and apply a construction of Serre in order to translate the formula in Theorem 5.1 into the language of quaternion algebras. The resulting quaternionic formulas in Corollary 7.15 are a model for the local intersection number calculation which is required for the computation of global height pairings in [GZ], except the condition rA (m) = 0 in Corollary 7.15 has to be dropped.
To avoid assuming rA (m) = 0, we have to confront the case of divisors which may contain components in common. Recall that global height pairings of degree 0 divisors are deﬁned in terms of local pairings, using a “moving lemma” to reduce to the case in which horizontal divisors intersect properly. We want an explicit formula for the global pairing c, Tm (dσ ) H where c = x − ∞ and d = x − 0, so we must avoid the abstract moving lemma and must work directly with improper intersections. Whereas [GZ] deal with this issue by using a technique from [Gr2, §5] which might be called “intersection theory with a tangent vector”, in §8 we develop a more systematic method which we call “intersection theory with a meromorphic tensor”. This theory is applied in §9 to give a formula in Theorem 9.6 which expresses a global height pairing in terms of local intersection numbers whose deﬁnition does not require proper intersections.
In §10 we use deformation theory to generalize Corollary 7.15 to include the case rA (m) 0, and Appendix A (by W.R. Mann) recovers all three main formulas in [GZ, Ch III, §9] as consequences of the quaternionic formulas we obtain via intersection theory with meromorphic tensors. The appendix follows the argument of Gross-Zagier quite closely, but explains some background, elaborates on some points in more detail than in [GZ, Ch III, §9], and works uniformly across all negative fundamental discriminants without parity restrictions. This parity issue is the main technical contribution of the appendix, and simply requires being a bit careful.
Some conventions. As in [GZ], we normalize the Artin map of class ﬁeld theory to associate uniformizers to arithmetic Frobenius elements. Thus, if K is an imaginary quadratic ﬁeld with Hilbert class ﬁeld H, then for a prime ideal p of K the isomorphism between Gal(H/K) and the class group ClK of K associates the ideal class [p] to an arithmetic Frobenius element.
Following [GZ], we only consider Heegner points with CM by the maximal order OK in an imaginary quadratic ﬁeld K. A Heegner diagram (for OK ) over an OK -scheme S is an OK -linear isogeny φ : E → E between elliptic curves over S which are equipped with OK -action which is “normalized” in the sense that the induced action on the tangent space at the identity is the same as that obtained through OS being a sheaf of OK -algebras via the OK -scheme structure on S. For example, when we speak of Heegner diagrams (or Heegner points on modular curves) over C, it is implicitly understood that an embedding K → C has been ﬁxed for all time.
Gal(H/K) on Heegner points X0 (N )(H) ⊆ X0 (N )(C), with N 1 having Under the action of ClK def all prime factors split in K, the action of [a] ∈ ClK sends the Heegner point ([b], n) = (C/b → C/n−1 b) to ([b][a]−1, n). The appearance of inversion is due to our decision to send uniformizers to arithmetic (rather than geometric) Frobenius elements. This analytic description of the Galois action on Heegner points will play a crucial role in the proof of Corollary 7.11.
If x is a Heegner point with associated CM ﬁeld K, we will write ux to denote the cardinality of the group √ √ of roots of unity in K (so ux = 2 unless K = Q( −1) or K = Q( −3)).
If S is a ﬁnite set, we will write either |S| or #S to denote the cardinality of S.
If (R, m) is a local ring, we write Rn to denote R/mn+1. If X and Y are R-schemes, we write HomRn (X, Y ) to denote the set of morphisms of mod mn+1 ﬁbers.
If B is a central simple algebra of ﬁnite dimension over a ﬁeld F, we write N : B → F and T : B → F to denote the reduced norm and reduced trace on B.
It is recommended (but not necessary) that anyone reading these notes should have a copy of [GZ] at hand. One piece of notation we adopt, following [GZ], is that rA (m) denotes the number of integral ideals of norm m 0 in an ideal class A of an imaginary quadratic ﬁeld K which is ﬁxed throughout the discussion.
Since rA = rA −1 due to complex conjugation inducing inversion on the class group without chaging norms,
GROSS-ZAGIER REVISITED 3if we change the Artin isomorphism Gal(H/K) ClK by a sign then this has no impact on a formula for c, Tm dσ v in terms of rA, where A ∈ ClK “corresponds” to σ ∈ Gal(H/K).
2. Some properties of abelian schemes and modular curves We begin by discussing some general facts about elliptic curves. Since the proofs for elliptic curves are the same as for abelian varieties, and more speciﬁcally it is not enough for us to work with passage between a number ﬁeld and C (e.g., we need to work over discrete valuation rings with positive characteristic residue ﬁeld, etc.) we state the basic theorem in the more natural setting of abelian varieties and abelian schemes.
Let A, B be abelian varieties over a ﬁeld F.
(1) If F is separably closed and F is an extension ﬁeld, then HomF (A, B) → HomF (A/F, B/F ) is an isomorphism. In other words, abelian varieties over a separably closed ﬁeld never acquire any “new” morphisms over an extension ﬁeld.
(2) If F = Frac(R) for a discrete valuation ring R and A, B have N´ron models A, B over R which e are proper (i.e., A and B have good reduction relative to R), then HomF (A, B) = HomR (A, B) → Homk (A0, B0 ) is injective, where k is the residue ﬁeld of R and (·)0 denotes the “closed ﬁber” functor on R-schemes.
In other words, (·)0 is a faithful functor from abelian schemes over R to abelian schemes over k. In fact, this latter faithfulness statement holds for abelian schemes over any local ring R whatsoever.
Proof. The technical details are a bit of a digression from the main aims of this paper, so although we do not know a reference we do not give the details. Instead, we mention the basic idea: for invertible on the base,
-power torsion is “relatively schematically dense” in an abelian scheme (in the sense of [EGA, IV3, 11.10]);
this ultimately comes down to the classical fact that such torsion is dense on an abelian variety over an algebraically closed ﬁeld. Such denseness, together with the fact that n -torsion is ﬁnite ´tale over the base, e provides enough rigidity to descend morphisms for the ﬁrst part of the theorem, and enough restrictiveness to force injectivity in the second part of the theorem.
We will later need the faithfulness of (·)0 in Theorem 2.1(2) for cases in which R is an artin local ring, so it is not adequate to work over discrete valuation rings and ﬁelds.
Let’s now recall the basic setup in [GZ]. We have ﬁxed an imaginary quadratic ﬁeld K ⊆ C with discriminant D 0 (and we take Q to be the algebraic closure of Q inside of C). We write H ⊆ Q for the Hilbert class ﬁeld of K, and we choose a positive integer N 1 which is relatively prime to D and for which all prime factors of N are split in K. Let X = X0 (N )/Z be the coarse moduli scheme as in [KM], so X is a proper ﬂat curve over Z which is smooth over Z[1/N ] but has some rather complicated ﬁbers modulo prime factors of N. We have no need for the assumption that D 0 is odd (i.e, D ≡ 1 mod 4), whose main purpose in [GZ] is to simplify certain aspects of calculations on the analytic side of the [GZ] paper, so we avoid such conditions (and hence include cases with even discriminant).
Let x ∈ X(H) ⊆ X(C) be a Heegner point with CM by the maximal order OK. Because of our explicit knowledge of the action of Gal(H/Q) on Heegner points, we see that the action of Gal(H/Q) on x ∈ X(H) is “free” (i.e., non-trivial elements in Gal(H/Q) do not ﬁx our Heegner point x ∈ X(H)), so it follows that the map x : Spec(H) → X/Q is a closed immersion. For each closed point v of Spec OH, by viewing H as the fraction ﬁeld of the algebraic localization OH,v we may use the valuative criterion for properness to uniquely extend x to a point in X(OH,v ). A simple “smearing out” argument involving denominator-chasing shows that all of these maps arise from a unique morphism x : Spec(OH ) → X.
It is not a priori clear if the map x is a closed immersion (in general it isn’t). More speciﬁcally, if Zx → X is the scheme-theoretic closure of x (i.e., the scheme-theoretic image of x), then Zx → Spec(Z) is proper, ﬂat, and quasi-ﬁnite, hence ﬁnite ﬂat, so it has the form Zx = Spec(Ax ) with Q ⊗Z Ax = H. Thus, Ax is an order in OH, but it isn’t obvious if Ax = OH. This is an important issue in subsequent intersection theory calculations because the intersection theory will involve closed subschemes of (various base chages on) X.
For this reason, we must distinguish Spec Ax → X and x : Spec OH → X.
4 BRIAN CONRADTo illustrate what can go wrong, consider the map Zp [T ] → Zp [ζp2 ] deﬁned by T → pζp2. Over Qp this deﬁnes an immersion φη : Spec(Qp (ζp2 )) → A1 p ⊆ P1 p which is a closed immersion since Qp (ζp2 ) = Z Z Qp [pζp2 ], and φη comes from a map φ : Spec(Zp [ζp2 ]) → P1 p which is necessarily the one we would get from Z applying the valuative criterion for properness to φη.
The “integral model” map φ is not a closed immersion because Zp [T ] → Zp [ζp2 ] deﬁned by T → pζp2 is not a surjection. In fact, the generic ﬁber closed subscheme of P1 p deﬁned by φη has scheme-theoretic Q closure in P1 p given by Spec(A) where A = image(φ) = Zp +pZp [ζp2 ] is a non-maximal order in Zp [ζp2 ]. The Z moral is that even if we can compute the “ﬁeld of deﬁnition” of a point on a generic ﬁber smooth curve, it is not true that the closure of this in a particular proper integral model of the curve is cut out by a Dedekind subscheme or that it lies in the relative smooth locus. It is a very fortunate fact in the Gross-Zagier situation that this diﬃculty usually does not arise for the “horizontal divisors” they need to consider.
Here is the basic result we need concerning Heegner divisors in the relative smooth locus.
Let x ∈ X(H) be a Heegner point and Λv denote the valuation ring of the completion Hv at a place v over a prime p. The map Spec(Λv ) → X/Zp corresponding to the pullback xv ∈ X(Hv ) of x factors through the relative smooth locus, and the induced natural map xv : Spec(Λv ) → X ×Z Λv arising from xv is a closed immersion into the relative smooth locus over Λv (and hence lies in the regular locus).